Questions tagged [density-operator]

The density operator describes a quantum system in an (in general mixed) state.

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12
votes
1answer
813 views

How can I write a Gaussian state as a squeezed, displaced thermal state

I would like to write a Gaussian state with density matrix $\rho$ (single mode) as a squeezed, displaced thermal state: \begin{gather} \rho = \hat{S}(\zeta) \hat{D}(\alpha) \rho_{\bar{n}} \hat{D}^\...
11
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0answers
273 views

How does one compute the state of a quantum system following imperfect measurement?

Suppose I have a quantum system $S$ ("system") with Hamiltonian $H_S$ and initial density matrix $\rho_S(0)$. I allow $S$ to interact with another system $P$ ("probe"), which has Hamiltonian $H_P$ and ...
8
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0answers
601 views

Green functions and density matrix

tl;dr The single particle density matrix is directly related to NEGF as shown here, I wish to find a way to relate NEGF also to density matrices which describe probability distribution of many body ...
7
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0answers
137 views

Completely positive maps and symmetric states

Let $\mathcal{N}$ be a completetely positive trace preserving map (aka a quantum channel) acting on a finite dimensional system $\mathrm{A}$, and let $\pi$ denote the maximally mixed state on $\mathrm{...
7
votes
1answer
460 views

Using open system dynamics to define a quantum state

Background The density matrix of a closed quantum system with Hilbert space $\mathscr H$ evolves according to the von Neumann equation \begin{align*} i\hbar\dot\rho=[H,\rho]. \end{align*} Given a ...
6
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1answer
342 views

Trying to understand mixed states

I took a basic quantum chemistry course (McQuarrie's "Quantum Chemistry"), but it never dealt with mixed states -- only pure states (or if it did, we never got to it in class). So I'm trying to ...
5
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0answers
825 views

Relation between Kraus operators and the Choi matrix

Let $\Phi$ be a CPTP map on density operators for a fixed $n-$dimensional state space and fix a basis $\{ | j\rangle \}$. I'm trying to understand the relationship between the Choi matrix $$M_\Phi:= \...
5
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0answers
140 views

What's the physical meaning of the eigenvalues of the spin-flipped density matrix?

In the computation of the entanglement of formation(EoF) of a 2 qubits mixed state, $\rho$, according to Wooters, we need to compute the concurrence of the state by computing the eigen values $\{\...
5
votes
1answer
436 views

How to understand Bloch sphere representation?

I'm really new to quantum computation. Now, I'm going through a tutorial article Quantum Computation: a Tutorial (NB: PDF). I was confused by certain points over there. So, on page 5, when the ...
4
votes
1answer
506 views

Solving the Lindblad quantum master equation in matrix form

I have just started learning density matrix and quantum master equations, and I am given a problem set that asks to find the solution to the Lindblad equation with $H$, $L_+$, $L_-$, $L_z$, and $\rho(...
4
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200 views

Steady state solution to density matrix

A density matrix follows the dynamics $$ \dot{\rho} = \mathcal{L}\rho, $$ where $\mathcal{L}$ is the Liouvillian super-operator. If put in Lindblad form, it can be written as $$ \mathcal{L}\rho = -...
4
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46 views

Aharonov-Bohm using density matrix?

On the one hand, we know that the overall phase of the wave function (of the whole system) is not a measurable quantity, but more an artifact of mathematical description — the physical states are rays ...
4
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0answers
472 views

Understanding quantum stochastic master equations

I'm teaching myself open quantum systems and the concept of a stochastic master equation has arisen. As someone who has studied classical stochastic processes a fair bit, this seems, at least to my ...
4
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1answer
252 views

Damped quantum harmonic oscillator - evolution of coherent state

I am trying to solve the following Master equation (also similar to damped quantum harmonic oscillator): $$\frac{d\hat{\rho}}{dt} = \frac{\Gamma}{2}\left(2\hat{a}\hat{\rho}\hat{a}^{\dagger} - \hat{a}^...
3
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0answers
17 views

Signal coherence/correlation vs quantum coherence

In general, I understand a signal $s(t) \in \mathbb{C}$ is called "coherent" when it has a large autocorrelation function. A pair of different signals $s(t)$, $r(t)$ can also be "coherent" if their ...
3
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0answers
42 views

Relation between maximally mixed state and thermal state

Hawking calculated the density matrix of the outgoing radiation to be a thermal state. I have heard people say this is a maximally mixed state. Is this because given a fixed average energy in the ...
3
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0answers
96 views

Open Quantum Systems: Born-Approximation and the preservation of Trace, Hermicity and Positivity

This is related to a previous question of mine. We consider a density matrix $\sigma(t)$ operating on a Hilbert space $\mathscr{H}_{s}\otimes \mathscr{H}_b$ with Hamiltonian $H = H_s \otimes \mathbb{...
3
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0answers
598 views

From the Heisenberg-Langevin equation to the Lindblad equation

In a open quantum system, one can easily derive the Heisenberg-Langevin equation of motion which describes the time evolution of creation/annihilation operators (in say, a cavity) $$\dot{a}(t) = i[H,...
3
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0answers
219 views

Entanglement of formation for two qutrits

For a pure bipartite state $|\psi\rangle\in\mathbb C^d\otimes \mathbb C^d$, the entanglement is given by $E(|\psi\rangle) = S(\mathrm{tr}_B|\psi\rangle\langle\psi|)$, with $S(\rho)=-\mathrm{tr}(\rho\...
3
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2answers
2k views

Reasoning Check: Trace of squared mixed-state density matrix

It's often written in the QI literature that, for a density operator $\rho$, if $\text{Tr}\left[\rho^{2}\right] < 1$, then $\rho$ describes a mixed state. However, I haven't seen any proofs of this ...
3
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1answer
214 views

Quantum master equation and off diagonal terms

I have a couple of related questions What is exactly the difference between the quantum master equation and the regular master equation? My understanding is that the normal master equation is used to ...
3
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0answers
386 views

Density matrix and entangled states

I am studying the density matrix formalism. I gather that: the trace of a density matrix, $tr(\rho)$ is always 1, if $tr(\rho^2) < 1$ we have a mixed state, otherwise a pure state, if $\rho$ ...
3
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0answers
208 views

Is there a physical significance to non-normal states of the algebra of observables?

Quantum theory may be formalized in several different ways. Generally, the physical discussion of different states of a quantum system distinguishes pure and mixed states, and then subsumes both in a ...
3
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0answers
308 views

What is the link between the density matrix and Hestenes' spinors in geometric algebra?

The density matrix (or state matrix) is a generalization of a wave function that is able to describe incoherent superpositions of an N-state system. It is often written as a matrix and observables are ...
2
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0answers
28 views

Expanding a density matrix in terms of operators

In Lukasz's paper: https://arxiv.org/pdf/0909.2654.pdf He writes "consider a density matrix ρ, written as a polynomial of the 2N Majoranas cj in such a way that each cj occurs to the power 0 or 1 in ...
2
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0answers
34 views

Diagonalisation of quasi-thermal state

I have the following density operator $$\frac{1}{t \pi N} \int_{\mathbb{C}} \mathrm{d}^2\gamma \exp \left[ -\frac{|\gamma+r\alpha|^2}{t^2 N} \right] |{\gamma}\rangle\langle{\gamma}|,$$ where $0\leq t,...
2
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1answer
156 views

Lindblad and Input-Output Formalism in Quantum Optics

I'm confused about how to apply the Lindblad formalism and the input-output formalism in practice, and how one goes between the two. Suppose I have a cavity (C) coupled to a reservoir (R), with the ...
2
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0answers
45 views

Expansion of an arbitrary density matrix in terms of coherent states?

It is well-known that any pure state can be expanded in terms of coherent states namely $$\left|\psi\right>=\frac{1}{\pi}\int d^2\alpha\left<\alpha|\psi\right>\left|\alpha\right>$$ due to ...
2
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0answers
110 views

How to calculate correlation functions for fermionic operators?

In the paper by Peschel (2003) https://arxiv.org/pdf/cond-mat/0212631.pdf How does one derive the following relation: $$ \langle c_{n}^\dagger c_{m}^\dagger c_{k}c_{l}\rangle = \langle c_{n}^\...
2
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0answers
28 views

Proof of factorization at late times for chaotic systems

While reading the paper "A bound on Chaos - Maldacena et. al", https://arxiv.org/abs/1503.01409 in equation (23) of the paper they factorize a correlator of the form, $$ Tr [\rho^{1/2} W(t) V \rho^{1/...
2
votes
1answer
104 views

Why do the density operators span the whole operator space $\mathcal{B}(H)$?

The convex set of density operators on a finite-dimensional Hilbert space $H$ defined by $\mathcal{D}(H):=\{\rho\in\mathcal{B}(H)|\,\rho\geq 0, \text{tr}\rho =1\}$ is said to span the entire space of ...
2
votes
1answer
88 views

Deriving a path-integral expression for a thermal density matrix with position-dependent temperature

I've been fiddling with deriving a path-integral expression for a thermal partition function with a position-dependent temperature but I'm not sure how to get started on this. Concretely, I'm trying ...
2
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0answers
49 views

What is the difference between complex and real coherneces?

If we have one density matrix $$\rho=\begin{bmatrix}1/2&&1/2\\1/2&&1/2\end{bmatrix}$$ for the state $\lvert\psi\rangle=\frac{1}{\sqrt2}(\lvert g\rangle+\lvert e\rangle)$ and a ...
2
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0answers
162 views

Reduced Density Matrix Fermions

Suppose we've got a fermionic state $$ \rho=\bigotimes_{i=1}^N|v_i\rangle\langle v_i| $$ where $|v_i\rangle=\left(\sqrt{d_i}c^\dagger(\psi_i)+\sqrt{1-d_i}c^\dagger(\eta_i)\right)|0\rangle_i$. The ...
2
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0answers
98 views

Solvable model with harmonic oscillator

I need help with a mathematical physics question. I have given the following system: A spin is coupled to a single harmonic oscillator mode with Hamiltonian $$H=(\epsilon/2) \sigma_{z} + \omega\, a^{...
2
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0answers
99 views

On Stinespring's Theorem

Wherein the proof of Stinespring's theorem does complete positivity enter? Suppose my map is not completely positive but the Kraus operators follow the trace preservation condition as $\sum_{n = 1}^{...
2
votes
0answers
129 views

Why density matrices in QFT are calculated only by going to Euclidean metric?

I have been reading a few papers on entanglement entropy. I noticed that whenever people calculate either the density matrix or the reduced density matrix of a specific region, it is usually done by ...
2
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0answers
62 views

quantum state homodyne tomography

I'm confused as to whether quantum state homodyne tomography can be successfully performed for detector efficiency less than 1/2, in the limit of infinite number of trials. My understanding is that ...
2
votes
0answers
138 views

Entanglement of bi- and tripartite pure and mixed states

since I'm not sure on how to find out whether a system is entangled or not I thought about examples that could clarify the whole thing. first example: system is in the state $\rho=1/2 (| 000 \rangle \...
2
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0answers
365 views

the density matrix in QFT on a cylinder

My question regards the density matrix in quantum field theory on a cylinder. The partition function is given by $Z=\text{Tr} e^{-\beta H}$. The elements of this thermal density matrix become \begin{...
2
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0answers
166 views

What are fragmented condensates?

It is defined that if more than one eigenvalue of the one-body density matrix are macroscopically occupied the condensate is said to be fragmented. $$ n^{(1)},n^{(2)},...=\mathcal{O}(\mathcal{N}) $$ ...
2
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0answers
476 views

What kind on transformations can be applied on density matrices?

Completely positive trace preserving maps ( CPTP ) transform a valid density matrix to another, then why do we only talk about unitary transformations on density matrices ( $\rho \to U\rho U^{\dagger}...
2
votes
0answers
83 views

Can I usefully interpret a non-unital completely positive (CP) map as a cooling process?

Non-unital completely positive (CP) maps take a maximally mixed quantum state (aka a normalized identity matrix aka an infinite temperature state) and map it to something else. This necessarily ...
2
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1answer
130 views

Density matrix in quantum computation and quantum statistical mechanics

What is the difference between the density matrix for quantum statistical mechanics and density matrix for quantum information theory?
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0answers
18 views

Have you ever seen: $\sqrt{\rho_{ee}\rho_{gg}}$-|$\rho_{eg}$|?

The next term appears in my research and it is quite meaningful: $\sqrt{\rho_{ee}\rho_{gg}}$-|$\rho_{eg}$| Where $\rho_{gg}$ and $\rho_{ee}$ are the populations in the excited and ground states, and $...
1
vote
0answers
23 views

Tracing $\rho (t)$ with respect to the Bath when system and bath are coupled in an open quantum system

Consider a system S that is coupled to a bath B. Let {$|s_i\rangle 's$} and {$|b_j\rangle 's$} be the eigen states of the system and bath hamiltonians respectively (i.e) \begin{align} \hat{H}_{S}|s_i\...
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0answers
45 views

Are the ideals in two GNS constructions linked to the equivalence (or not) of the CCR representations?

Starting from the abstract C* algebra $A$ of canonical commutative relations, a state $\rho$ over this algebra enables to construct a Hilbert space $A/I$ where $I$ is the ideal of the elements $a$ ...
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0answers
47 views

Quantum Coherence in a Two-level System in the Density Matrix Formalism

Dealing with semiclassical light-matter interaction, in particular the interaction between an electromagnetic field and a two level system using the density matrix formalism, I learned that the system ...
1
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0answers
51 views

Can we write the Quantum Fidelity between two density operators in terms of Quasi-Probability Distributions: $P$, $Q$ and $W$?

Quantum Fidelity between two density operators, $\hat{\rho}$ and $\hat{\sigma}$, is given by $F(\hat{\rho},\hat{\sigma})=\left(Tr\sqrt{\sqrt{\hat{\rho}}\hat{\sigma}\sqrt{\hat{\rho}}}\right)^2$, where $...
1
vote
1answer
91 views

reduced density matrix of state

given a multi particle state I have to calculate the reduced density matrix where I trace out the third particle for this I first calculate the corresponding 2D density matrix with the bra vector of ...